where \((\mathbf {K})_{ij} = \varphi (\Vert \mathbf {x}_i - \mathbf {x}_j\Vert )\) , \((\mathbf {P})_{ij} = p_j(\mathbf {x}_i)\)  and \(\sigma \)  is a smoothing parameter that eventually relaxes the interpolation condition in order to smoothen the “interpolant” function. For the specific case has been set to \(\sigma = 0\) . Selecting a RBF which is positive definite of order \(q\)  with \(\mathbf {P}\)  that has full column rank, the solution is unique provided that the degree of the monomial terms is at least \(m=q-1\)  [1]}, [2]}. In the specific case, for each snapshot and solution field, the interpolant function has been constructed using observation only on the active nodes. The solution field on the inactive nodes has been reconstructed using the previously defined interpolant function:
\(U(\mathbf {x}_{i_{\text{inactive}}}) = U_I(\mathbf {x}_{i_{\text{inactive}}}).\) 
